Well-Posedness Results for the Three-Dimensional Zakharov-Kuznetsov Equation

We prove the local well-posedness of the three-dimensional Zakharov--Kuznetsov equation $\partial_tu+\Delta\partial_xu+ u\partial_xu=0$ in the Sobolev spaces $H^s(\mathbb{R}^3)$, $s>1$, as well as in the Besov space $B^{1,1}_2(\mathbb{R}^3)$. The proof is based on a sharp maximal function estimate in time-weighted spaces.